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Asymptotic expansion for harmonic functions in the half-space with a pressurized cavity. (English) Zbl 1350.35058
Summary: In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half-space with a cavity $$C$$. Zero normal derivative is assumed at the boundary of the half-space; differently, at $$\partial C$$, the normal derivative of the function is required to be given by an external datum $$g$$, corresponding to a pressure term exerted on the medium at $$\partial C$$. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half-space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum $$g$$, which describes a constant pressure at $$\partial C$$, we recover a simplified representation based on a polarization tensor.

##### MSC:
 35C20 Asymptotic expansions of solutions to PDEs 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 35J25 Boundary value problems for second-order elliptic equations 35Q86 PDEs in connection with geophysics
##### Keywords:
single and double layer potentials; volcanology
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